The 3d organization of the genome - in particular, which two regions of DNA are in contact with each other - plays a role in regulating gene expression. Several factors influence genome 3d organization. Nucleosomes (where ~ 100 basepairs of DNA wrap around histone proteins) also bend, twist and compactify chromosomal DNA, altering its polymer mechanics. How much does the positioning of nucleosomes between gene loci influence contacts between those gene loci? And, to what extent is polymer mechanics responsible for this? To address this question, we combine a stochastic polymer mechanics model of chromosomal DNA including twists and wrapping induced by nucleosomes with two data-driven pipelines. The first estimates nucleosome positioning from ATACseq data in regions of high accessibility. Most of the genome is low-accessibility, so we combine this with a novel image analysis method that estimates the distribution of nucleosome spacing from electron microscopy data. There are no free parameters in the biophysical model. We apply this method to IL6, IL15, CXCL9, and CXCL10, inflammatory marker genes in macrophages, before and after immune stimulation, and compare the predictions with contacts measured by conformation capture experiments (4C-seq). We find that within a 500 kilo-basepairs genomic region, polymer mechanics with nucleosomes can explain 71% of close contacts. These results suggest that, while genome contacts on 100kbp-scales are multifactorial, they may be amenable to mechanistic, physical explanation. Our work also highlights the role of nucleosomes, not just at the loci of interest, but between them, and not just the total number of nucleosomes, but their specific placement. The method generalizes to other genes, and can be used to address whether a contact is under active regulation by the cell (e.g., a macrophage during inflammatory stimulation). Importantly, our findings suggest that gene function may have evolved through selective pressures that co-opted contact-mediated regulatory mechanisms reliant largely on polymer mechanics.